Suppose that a body of mass m moving at speed u collides with a stationary body of mass $2m$ and sticks to it, so the coefficient of restitution ($e$) is zero. Applying conservation of momentum to determine the speed of the composite body after the collision, you'll find that 1/3 of the initial kinetic energy has been retained. If we define a perfectly inelastic collision as one in which all the kinetic energy is lost, then the collision is not perfectly inelastic even though $e=0$. This, presumably, is what the Wiki claim means.
I have issues with the supposed definition just given. If we look at the collision in the frame of reference of the centre of mass of the system, then both bodies come to rest when they stick together, and all the kinetic energy is lost. So in this frame, we'd have to say that the collision is perfectly inelastic. So, with the supposed definition given above, whether or not a collision is perfectly inelastic would depend both on the collision itself and on the frame of reference in which we view it. Unsatisfactory!
We therefore have a choice: either don't use the term 'perfectly inelastic', or use it to mean that in the centre of mass frame all kinetic energy is lost – which happens if and only if $e=0$. In the first case the Wiki claim is meaningless, in the second case it is false.
There is no similar issue with perfectly elastic collisions. If kinetic energy is conserved in one frame of reference, it is conserved in all.